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Here is the problem: A sample of 36 has mean 100; the population standard deviation is 15. Find the 95 percent confidence interval for the mean.

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Build a 95 percent confidence interval for the population mean μ.

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Standard error of the mean.

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Critical value for 95 percent confidence. The middle 95 percent leaves 2.5 percent in each tail. σ is treated as known, so the standard normal distribution gives the critical value.

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Margin of error.

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Interval: estimate plus or minus margin.

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So the answer is 95 percent CI: (95.1001, 104.9); margin of error 4.89991.

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We are 95 percent confident that the population mean μ lies between 95.1001 and 104.9. The confidence describes the method: about 95 percent of intervals built this way from repeated samples would capture the true value.

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The interval is centred on the estimate: (95.1001 plus 104.9) / 2 equals 100.